Using first-principles calculations, we study the growth process of bismuth (Bi) films on Cu (111) substrate. By analyzing the formation mechanism of single layer alpha-bismuthene and stability of multilayered Bi layers, we reveal that Bi atoms show quasi-one-dimensional growth model by developing periodic zigzag chains, and eventually form a Bi atomic layer on Cu (111) surface. Flat surface alpha-bismuthene (F alpha-Bi) can be formed by depositing two Bi atomic layers on Cu (111) and maintains its stability below 60 K. In the multilayered Bi atomic layers, the stability of Bi films shows an oscillatory behavior. It is more stable for Bi films with even layers. The charge density between even and odd films shows that the stability is associated with the saturation of p(z) orbital of Bi atoms. Besides, the distorted surface alpha-bismuthene (D alpha-Bi) can maintain its stability up to 450 K on Cu (111) surface. Further study shows that the Cu substrate strongly affects the properties of F alpha-Bi than that of D alpha-Bi. Our studies provide guidance for fabricating stable Bi (110) thin films based devices.